↑

Properties

Label 2070.2.j.h
Level $2070$
Weight $2$
Character orbit 2070.j
Analytic conductor $16.529$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2070,2,Mod(323,2070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2070, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2070.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.j (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 24 x^{14} - 48 x^{13} + 160 x^{12} - 292 x^{11} + 436 x^{10} - 176 x^{9} - 914 x^{8} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} - \beta_{7} q^{4} + (\beta_{10} - \beta_{4}) q^{5} - \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} - \beta_{7} q^{4} + (\beta_{10} - \beta_{4}) q^{5} - \beta_{4} q^{8} + ( - \beta_{12} - 1) q^{10} + (\beta_{14} + \beta_{13} + \cdots + \beta_{4}) q^{11}+ \cdots - 7 \beta_{4} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{10} - 16 q^{13} - 16 q^{16} + 8 q^{22} - 16 q^{25} - 96 q^{31} + 24 q^{37} + 8 q^{43} - 16 q^{46} + 16 q^{52} - 32 q^{58} + 16 q^{61} - 8 q^{67} - 32 q^{73} + 16 q^{76} + 32 q^{82} + 96 q^{85} + 8 q^{88} + 80 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 24 x^{14} - 48 x^{13} + 160 x^{12} - 292 x^{11} + 436 x^{10} - 176 x^{9} - 914 x^{8} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 52\!\cdots\!04 \nu^{15} + \cdots - 84\!\cdots\!67 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22\!\cdots\!78 \nu^{15} + \cdots - 17\!\cdots\!01 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 53\!\cdots\!70 \nu^{15} + \cdots - 45\!\cdots\!11 ) / 19\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 57\!\cdots\!25 \nu^{15} + \cdots - 69\!\cdots\!60 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 24\!\cdots\!32 \nu^{15} + \cdots + 15\!\cdots\!95 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 76\!\cdots\!09 \nu^{15} + \cdots - 29\!\cdots\!38 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 20548177138859 \nu^{15} + 90486844173851 \nu^{14} - 526301101065768 \nu^{13} + \cdots - 46\!\cdots\!96 ) / 25\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 40\!\cdots\!23 \nu^{15} + \cdots - 75\!\cdots\!99 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 63\!\cdots\!12 \nu^{15} + \cdots + 32\!\cdots\!51 ) / 69\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 58\!\cdots\!59 \nu^{15} + \cdots + 18\!\cdots\!03 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 20\!\cdots\!96 \nu^{15} + \cdots - 65\!\cdots\!23 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 22\!\cdots\!16 \nu^{15} + \cdots - 42\!\cdots\!83 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 25\!\cdots\!45 \nu^{15} + \cdots - 42\!\cdots\!56 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 27\!\cdots\!97 \nu^{15} + \cdots - 78\!\cdots\!50 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 89\!\cdots\!37 \nu^{15} + \cdots - 17\!\cdots\!57 ) / 34\!\cdots\!95 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{12} - \beta_{7} + \beta_{5} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} + \beta_{11} - 2\beta_{8} - 4\beta_{7} - 2\beta_{6} + 8\beta_{4} - \beta_{3} - \beta_{2} + 2\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - 2 \beta_{14} + \beta_{13} - 7 \beta_{12} + 3 \beta_{11} - 8 \beta_{10} - 2 \beta_{9} + \cdots - 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - 12 \beta_{12} - 12 \beta_{11} - 8 \beta_{10} + 8 \beta_{8} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 15 \beta_{15} + 15 \beta_{14} - 35 \beta_{13} - 3 \beta_{12} - 101 \beta_{11} + 66 \beta_{10} + \cdots + 116 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 84 \beta_{15} + 120 \beta_{14} + 231 \beta_{12} + 133 \beta_{11} + 320 \beta_{10} + 84 \beta_{9} + \cdots + 805 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 516 \beta_{15} + 215 \beta_{14} + 516 \beta_{13} + 989 \beta_{12} + 1434 \beta_{11} + 184 \beta_{10} + \cdots + 184 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 672 \beta_{14} + 672 \beta_{13} - 136 \beta_{12} + 1088 \beta_{11} - 1656 \beta_{10} - 952 \beta_{9} + \cdots - 5287 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 7298 \beta_{15} - 7298 \beta_{14} - 3026 \beta_{13} - 16461 \beta_{12} - 11140 \beta_{11} - 14166 \beta_{10} + \cdots - 28531 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 19844 \beta_{15} - 28072 \beta_{13} - 39975 \beta_{12} - 67035 \beta_{11} + 5104 \beta_{10} + \cdots + 56293 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 42311 \beta_{15} + 102130 \beta_{14} - 42311 \beta_{13} + 119839 \beta_{12} - 45011 \beta_{11} + \cdots + 543269 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 201069 \beta_{15} + 142170 \beta_{14} + 142170 \beta_{13} + 431244 \beta_{12} + 431244 \beta_{11} + \cdots + 431244 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 590931 \beta_{15} - 590931 \beta_{14} + 1426591 \beta_{13} + 848263 \beta_{12} + 3051843 \beta_{11} + \cdots - 5201544 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 4023804 \beta_{15} - 5690568 \beta_{14} - 9424009 \beta_{12} - 3630171 \beta_{11} - 12076064 \beta_{10} + \cdots - 27456559 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 19935288 \beta_{15} - 8257509 \beta_{14} - 19935288 \beta_{13} - 39675681 \beta_{12} - 52148388 \beta_{11} + \cdots - 4206130 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(1891\)
\(\chi(n)\) \(-1\) \(\beta_{7}\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1
−1.03797 − 2.50588i
−0.283623 − 0.684727i
0.889432 + 2.14728i
1.43216 + 3.45754i
1.43216 − 0.593221i
0.889432 − 0.368415i
−0.283623 + 0.117481i
−1.03797 + 0.429941i
−1.03797 + 2.50588i
−0.283623 + 0.684727i
0.889432 − 2.14728i
1.43216 − 3.45754i
1.43216 + 0.593221i
0.889432 + 0.368415i
−0.283623 − 0.117481i
−1.03797 − 0.429941i
−0.707107 − 0.707107i 0 1.00000i −1.46791 + 1.68678i 0 0 0.707107 − 0.707107i 0 2.23071 − 0.154765i
323.2 −0.707107 − 0.707107i 0 1.00000i −0.401104 − 2.19980i 0 0 0.707107 − 0.707107i 0 −1.27187 + 1.83912i
323.3 −0.707107 − 0.707107i 0 1.00000i 1.25785 − 1.84873i 0 0 0.707107 − 0.707107i 0 −2.19669 + 0.417821i
323.4 −0.707107 − 0.707107i 0 1.00000i 2.02538 + 0.947538i 0 0 0.707107 − 0.707107i 0 −0.762151 − 2.10217i
323.5 0.707107 + 0.707107i 0 1.00000i −2.02538 − 0.947538i 0 0 −0.707107 + 0.707107i 0 −0.762151 − 2.10217i
323.6 0.707107 + 0.707107i 0 1.00000i −1.25785 + 1.84873i 0 0 −0.707107 + 0.707107i 0 −2.19669 + 0.417821i
323.7 0.707107 + 0.707107i 0 1.00000i 0.401104 + 2.19980i 0 0 −0.707107 + 0.707107i 0 −1.27187 + 1.83912i
323.8 0.707107 + 0.707107i 0 1.00000i 1.46791 − 1.68678i 0 0 −0.707107 + 0.707107i 0 2.23071 − 0.154765i
737.1 −0.707107 + 0.707107i 0 − 1.00000i −1.46791 − 1.68678i 0 0 0.707107 + 0.707107i 0 2.23071 + 0.154765i
737.2 −0.707107 + 0.707107i 0 − 1.00000i −0.401104 + 2.19980i 0 0 0.707107 + 0.707107i 0 −1.27187 − 1.83912i
737.3 −0.707107 + 0.707107i 0 − 1.00000i 1.25785 + 1.84873i 0 0 0.707107 + 0.707107i 0 −2.19669 − 0.417821i
737.4 −0.707107 + 0.707107i 0 − 1.00000i 2.02538 − 0.947538i 0 0 0.707107 + 0.707107i 0 −0.762151 + 2.10217i
737.5 0.707107 − 0.707107i 0 − 1.00000i −2.02538 + 0.947538i 0 0 −0.707107 − 0.707107i 0 −0.762151 + 2.10217i
737.6 0.707107 − 0.707107i 0 − 1.00000i −1.25785 − 1.84873i 0 0 −0.707107 − 0.707107i 0 −2.19669 − 0.417821i
737.7 0.707107 − 0.707107i 0 − 1.00000i 0.401104 − 2.19980i 0 0 −0.707107 − 0.707107i 0 −1.27187 − 1.83912i
737.8 0.707107 − 0.707107i 0 − 1.00000i 1.46791 + 1.68678i 0 0 −0.707107 − 0.707107i 0 2.23071 + 0.154765i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2070.2.j.h ✓ 16
3.b odd 2 1 inner 2070.2.j.h ✓ 16
5.c odd 4 1 inner 2070.2.j.h ✓ 16
15.e even 4 1 inner 2070.2.j.h ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2070.2.j.h ✓ 16 1.a even 1 1 trivial
2070.2.j.h ✓ 16 3.b odd 2 1 inner
2070.2.j.h ✓ 16 5.c odd 4 1 inner
2070.2.j.h ✓ 16 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2070, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{8} + 96T_{11}^{6} + 2976T_{11}^{4} + 30400T_{11}^{2} + 40000 \) Copy content Toggle raw display
\( T_{17}^{16} + 3008T_{17}^{12} + 2143744T_{17}^{8} + 519159808T_{17}^{4} + 34828517376 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 8 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 96 T^{6} + \cdots + 40000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 8 T^{7} + \cdots + 92416)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 34828517376 \) Copy content Toggle raw display
$19$ \( (T^{8} + 144 T^{6} + \cdots + 51984)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} - 160 T^{6} + \cdots + 614656)^{2} \) Copy content Toggle raw display
$31$ \( (T + 6)^{16} \) Copy content Toggle raw display
$37$ \( (T^{8} - 12 T^{7} + \cdots + 254016)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 168 T^{6} + \cdots + 535824)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 4 T^{7} + \cdots + 287296)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 33024 T^{12} + \cdots + 16777216 \) Copy content Toggle raw display
$53$ \( T^{16} + 4976 T^{12} + \cdots + 1679616 \) Copy content Toggle raw display
$59$ \( (T^{8} - 176 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 4 T^{3} + \cdots + 3940)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 4 T^{7} + \cdots + 29160000)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 376 T^{6} + \cdots + 8503056)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 16 T^{7} + \cdots + 501264)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 96 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 385571451136 \) Copy content Toggle raw display
$89$ \( (T^{8} - 160 T^{6} + \cdots + 614656)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 40 T^{7} + \cdots + 137545984)^{2} \) Copy content Toggle raw display
show more
show less